Ukrainian mathematical congress  2009
Viacheslav Shtyk (M. M. Bogolyubov Institute for Theoretical Physics, Kyiv, Ukraine) Meanfield limit for the dynamics of quantum particle systems We will introduce the meanfield asymptotic of a solution of the initialvalue problem of the BBGKY hierarchy of quantum manyparticle systems. We construct a solution as an expansion over particle clusters whose evolution are governed by the correspondingorder cumulant (semiinvariant) of the evolution operators of finitely many particles. We prove that in the meanfield limit the constructed solution converges to the sequence of marginal operators satisfying the corresponding initialvalue problem of the quantum Vlasov hierarchy in the sense of the norm convergence of the space of sequences of trace class operators. The solution of the initialvalue problem of the limiting hierarchy possesses of the chaos property which make it possible to substantiate the derivation of the suitable nonlinear kinetic equation  quantum Vlasov equation and as the consequece  the nonlinear Schrödinger equation.
